This painful question is inspired by the question 
[non-Lie subgroups](http://mathoverflow.net/questions/3157/non-lie-subgroups) . Let R denote the real numbers. Let f be an discontinuous additive map from R --> R. Is it possible that the graph of f, inside R^2 with the usual topology, is connected? Remember that the graph of discontinuous function can be connected, as in the 
[Toplogist's sine curve](http://en.wikipedia.org/wiki/Topologist%27s_sine_curve).)

I was hoping to show that any connected subgroup of a Lie group is a Lie group, thus answering the previous question, and I couldn't get rid of this horrible example.